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Measure-Independent Characterization of Contrast Optimal Visual Cryptography Schemes

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Information Theoretic Security (ICITS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8317))

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Abstract

The contrast in visual cryptography has received a lot of attention. It has been studied using three different measures. In this paper we follow a measure-independent approach, which, by using the structural properties of the schemes, enables us to provide a characterization of optimal schemes that is independent of the specific measure used to assess the contrast. In particular we characterize and provide constructions of optimal schemes for the cases of \((2,n)\)-threshold and \((n,n)\)-threshold schemes. Then, we apply the measure-independent results to the three measures that have been used in the literature obtaining both new characterizations and constructions of optimal schemes and alternative proofs of known results.

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Notes

  1. 1.

    For deterministic visual cryptography it must be \(m\ge 2\), i.e., the pixel expansion is unavoidable. The probabilistic and the random grid visual cryptography models allow \(m=1\).

  2. 2.

    In a more general form, it is possible to consider access structures where there are some subsets that are neither qualified nor forbidden; in such a case we simply don’t care about what those subsets of participants can do with the shares.

  3. 3.

    The authors of [32] used two thresholds, \(\hat{\ell }\) and \(\hat{h}\), to measure the level of whiteness in the reconstruction of a white or a black pixel, while we use two thresholds, \(\ell \) (low) and \(h\) (high), to measure the level of blackness in the reconstruction of a white or a black pixel. In [21] the thresholds measure the level of blackness too, but are expressed as \(m-\hat{h}\) and \(m-\hat{\ell }\). In the first paper on visual cryptography [28] explicit thresholds \(\ell \) and \(h\) are not used, but the conditions are stated in terms of the level of blackness, too.

  4. 4.

    According to the measure \(\gamma _{\textsc {ns}}\) the contrast in both cases is the same.

  5. 5.

    We state the following theorem using the original contrast thresholds \(\hat{\ell }=m-h\), \(\hat{h}=m-\ell \) as they appear in the original paper.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Salerno, via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Paolo D’Arco, Roberto De Prisco & Alfredo De Santis

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  1. Paolo D’Arco
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  2. Roberto De Prisco
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  3. Alfredo De Santis
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Correspondence to Roberto De Prisco .

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  1. Nanyang Technological University, Singapore, Singapore

    Carles Padró

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D’Arco, P., De Prisco, R., De Santis, A. (2014). Measure-Independent Characterization of Contrast Optimal Visual Cryptography Schemes. In: Padró, C. (eds) Information Theoretic Security. ICITS 2013. Lecture Notes in Computer Science(), vol 8317. Springer, Cham. https://doi.org/10.1007/978-3-319-04268-8_3

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  • DOI: https://doi.org/10.1007/978-3-319-04268-8_3

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